In mechanics, strain is defined as relative deformation, compared to a reference position configuration.

Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the final configuration of the body and on whether the metric tensor or its dual is considered.

Hence strains are dimensionless and are usually expressed as a decimal fraction or a percentage.

The spatial derivative of a uniform translation is zero, thus strains measure how much a given displacement differs locally from a rigid-body motion.

The amount of stretch or compression along material line elements or fibers is the normal strain, and the amount of distortion associated with the sliding of plane layers over each other is the shear strain, within a deforming body.

[2] This could be applied by elongation, shortening, or volume changes, or angular distortion.

[3] The state of strain at a material point of a continuum body is defined as the totality of all the changes in length of material lines or fibers, the normal strain, which pass through that point and also the totality of all the changes in the angle between pairs of lines initially perpendicular to each other, the shear strain, radiating from this point.

However, it is sufficient to know the normal and shear components of strain on a set of three mutually perpendicular directions.

Engineering strain, also known as Cauchy strain, is expressed as the ratio of total deformation to the initial dimension of the material body on which forces are applied.

In the case of a material line element or fiber axially loaded, its elongation gives rise to an engineering normal strain or engineering extensional strain e, which equals the relative elongation or the change in length ΔL per unit of the original length L of the line element or fibers (in meters per meter).

The normal strain is positive if the material fibers are stretched and negative if they are compressed.

The true shear strain is defined as the change in the angle (in radians) between two material line elements initially perpendicular to each other in the undeformed or initial configuration.

The engineering shear strain is defined as the tangent of that angle, and is equal to the length of deformation at its maximum divided by the perpendicular length in the plane of force application, which sometimes makes it easier to calculate.

The stretch ratio or extension ratio (symbol λ) is an alternative measure related to the extensional or normal strain of an axially loaded differential line element.

The extension ratio λ is related to the engineering strain e by

This equation implies that when the normal strain is zero, so that there is no deformation, the stretch ratio is equal to unity.

The stretch ratio is used in the analysis of materials that exhibit large deformations, such as elastomers, which can sustain stretch ratios of 3 or 4 before they fail.

On the other hand, traditional engineering materials, such as concrete or steel, fail at much lower stretch ratios.

) is defined in the International System of Quantities (ISQ), more specifically in ISO 80000-4 (Mechanics), as a "tensor quantity representing the deformation of matter caused by stress.

"[6] ISO 80000-4 further defines linear strain as the "quotient of change in length of an object and its length" and shear strain as the "quotient of parallel displacement of two surfaces of a layer and the thickness of the layer".

The strain tensor can then be expressed in terms of normal and shear components as:

Consider a two-dimensional, infinitesimal, rectangular material element with dimensions dx × dy, which, after deformation, takes the form of a rhombus.

For very small displacement gradients the squares of the derivative of

The normal strain in the x-direction of the rectangular element is defined by

Similarly, the normal strain in the y- and z-directions becomes

The engineering shear strain (γxy) is defined as the change in angle between lines AC and AB.

By interchanging x and y and ux and uy, it can be shown that γxy = γyx.

Actually, if we consider a cube with an edge length a, it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions

A strain field associated with a displacement is defined, at any point, by the change in length of the tangent vectors representing the speeds of arbitrarily parametrized curves passing through that point.

A basic geometric result, due to Fréchet, von Neumann and Jordan, states that, if the lengths of the tangent vectors fulfil the axioms of a norm and the parallelogram law, then the length of a vector is the square root of the value of the quadratic form associated, by the polarization formula, with a positive definite bilinear map called the metric tensor.